PARTIAL REGULARITY FOR OPTIMAL TRANSPORT MAPS
GUIDO DE PHILIPPIS AND ALESSIO FIGALLI
Abstract. We prove that, for general cost functions on Rn , or for the cost d2 /2 on a Riemannian
manifold, optimal transport maps between smooth densities are always smooth outside a closed
singular set of measure zero.
1. Introduction
A natural and important issue in optimal transport theory is the regularity of optimal transport
maps. Indeed, apart from being a typical PDE/analysis question, knowing whether optimal maps
are smooth or not is an important step towards a qualitative understanding of them.
It is by now well known that, for the smoothness of optimal maps, conditions on both the cost
function and on the geometry of the supports of the measures are needed.
In the special case c(x, y) = |x − y|2 /2 on Rn , Caffarelli [3, 4, 5, 6] proved regularity of optimal
maps under suitable assumptions on the densities and on the geometry of their support. More
precisely, in its simplest form, Caffarelli’s result states as follows:
Theorem 1.1. Let f and g be smooth probability densities, respectively bounded away from zero
and infinity on two bounded open sets X and Y , and let T : X → Y denote the unique optimal
transport map from f to g for the quadratic cost |x − y|2 /2. If Y is convex, then T is smooth inside
X. On the other hand, if Y is not convex, then there exist smooth densities f and g (both bounded
away from zero and infinity on X and Y , respectively) for which the map T is not continuous.
A natural question which arises from the previous result is whether one may prove some partial
regularity on T when the convexity assumption on Y is removed. In [16, 18] the authors proved
the following result:
Theorem 1.2. Let f and g be smooth probability densities, respectively bounded away from zero
and infinity on two bounded open sets X and Y , and let T : X → Y denote the unique optimal
transport map from f to g for the quadratic cost |x − y|2 /2. Then there exist two open sets X 0 ⊂ X
and Y 0 ⊂ Y , with |X \ X 0 | = |Y \ Y 0 | = 0, such that T : X 0 → Y 0 is a smooth diffeomorphism.
In the case of general cost functions on Rn , or when c(x, y) = d(x, y)2 /2 on a Riemannian
manifold M (d(x, y) being the Riemannian distance), the situation is much more complicated.
Indeed, as shown by Ma, Trudinger, and Wang [33], and Loeper [31], in addition to suitable
convexity assumptions on the support of the target density (or on the cut locus of the manifold
when supp(g) = M [24]), a very strong structural condition on the cost function, the so-called
MTW condition, is needed to ensure the smoothness of the map.
More precisely, if the MTW condition holds (together with some suitable convexity assumptions
on the target domain), then the optimal map is smooth [35, 36, 21, 30, 19]. On the other hand, if
the MTW condition fails at one point, then one can construct smooth densities (both supported
1
2
G. DE PHILIPPIS AND A. FIGALLI
on domains which satisfy the needed convexity assumptions) for which the optimal transport map
is not continuous [31] (see also [15]).
In the case of Riemannian manifolds, the MTW condition for c = d2 /2 is very restrictive: indeed,
as shown by Loeper [31], it implies that M has non-negative sectional curvature, and actually it is
much stronger than the latter [28, 23]. In particular, if M has negative sectional curvature, then
the MTW condition fails at every point. Let us also mention that, up to now, the MTW condition
is known to be satisfied only for very special classes of Riemannian manifolds, such as spheres, their
products, their quotients and submersions, and their perturbations [32, 22, 10, 29, 25, 20, 11], and
for instance it is known to fail on sufficiently flat ellipsoids [23].
The goal of the present paper is to show that, even without any condition on the cost function
or on the supports of the densities, optimal transport maps are always smooth outside a closed
singular set of measure zero. In order to state our results, we first have to introduce some basic
assumptions on the cost functions which are needed to ensure existence and uniqueness of optimal
maps. As before, X and Y denote two open subsets of Rn .
(C0)
(C1)
(C2)
(C3)
The cost function c : X × Y → R is of class C 2 with kckC 2 (X×Y ) < ∞.
For any x ∈ X, the map Y 3 y 7→ −Dx c(x, y) ∈ Rn is injective.
For any y ∈ Y , the map X 3 x 7→ −Dy c(x, y) ∈ Rn is injective.
det(Dxy c)(x, y) 6= 0 for all (x, y) ∈ X × Y .
Here are our main results:
Theorem 1.3. Let X, Y ⊂ Rn be two bounded open sets, and let f : X → R+ and g : Y → R+
be two continuous probability densities, respectively bounded away from zero and infinity on X
and Y . Assume that the cost c : X × Y → R satisfies (C0)-(C3), and denote by T : X → Y
the unique optimal transport map sending f onto g. Then there exist two relatively closed sets
ΣX ⊂ X, ΣY ⊂ Y of measure zero such that T : X \ ΣX → Y \ ΣY is a homeomorphism of class
0,β
k+2,α
k,α
k,α
Cloc
for any β < 1. In addition, if c ∈ Cloc
(X × Y ), f ∈ Cloc
(X), and g ∈ Cloc
(Y ) for some
k+1,α
k ≥ 0 and α ∈ (0, 1), then T : X \ ΣX → Y \ ΣY is a diffeomorphism of class Cloc .
Theorem 1.4. Let M be a smooth Riemannian manifold, and let f, g : M → R+ be two continuous
probability densities, locally bounded away from zero and infinity on M . Let T : M → M denote
the optimal transport map for the cost c = d2 /2 sending f onto g. Then there exist two closed sets
0,β
ΣX , ΣY ⊂ M of measure zero such that T : M \ ΣX → M \ ΣY is a homeomorphism of class Cloc
for any β < 1. In addition, if both f and g are of class C k,α , then T : M \ ΣX → M \ ΣY is a
k+1,α
diffeomorphism of class Cloc
.
The paper is structured as follows: in the next section we introduce some notation and preliminary results. Then, in Section 3, we show how both Theorem 1.3 and Theorem 1.4 are a direct
consequence of some local regularity results around differentiability points of T , see Theorems 4.3
and 5.3. Finally, Sections 4 and 5 are devoted to the proof of these local results.
Acknowledgements: We wish to thank Luigi Ambrosio for his careful reading of a preliminary
version of this manuscript. AF is partially supported by NSF Grant DMS-0969962. Both authors
acknowledge the support of the ERC ADG Grant GeMeThNES.
PARTIAL REGULARITY FOR OPTIMAL TRANSPORT MAPS
3
2. Notation and preliminary results
Through a well established procedure, maps that solve optimal transport problems derive from
a c-convex potential, itself solution to a Monge-Ampère type equation.
More precisely, given a cost function c : X × Y → R, a function u : X → R is said c-convex if it
can be written as
(2.1)
u(x) = sup {−c(x, y) + λy } ,
y∈Y
for some constants λy ∈ R ∪ {−∞}.
Similarly to the subdifferential for convex function, for c-convex functions one can talk about
their c-subdifferential: if u : X → R is a c-convex function as above, the c-subdifferential of u at x
is the (nonempty) set
(2.2)
∂c u(x) := {y ∈ Y : u(z) ≥ −c(z, y) + c(x, y) + u(x)
∀ z ∈ X}.
If x0 ∈ X and y0 ∈ ∂c u(x0 ), we will say that the function
(2.3)
Cx0 ,y0 (·) := −c(·, y0 ) + c(x0 , y0 ) + u(x0 )
is a c-support for u at x0 . We also define the Frechet subdifferential of u at x as
∂ − u(x) := p ∈ Rn : u(z) ≥ u(x) + p · (z − x) + o(|z − x|) .
We will use the following notation: if E ⊂ X then
[
[
∂c u(E) :=
∂c u(x),
∂ − u(E) :=
∂ − u(x).
x∈E
It is easy to check that, if c is of class
(2.4)
x∈E
C 1,
y ∈ ∂c u(x)
then the following inclusion holds:
=⇒
−Dx c(x, y) ∈ ∂ − u(x).
In addition, if c satisfies (C0)-(C2), then we can define the c-exponential map:
c-expx (p) = y ⇔ p = −Dx c(x, y)
n
(2.5)
for any x ∈ X, y ∈ Y , p ∈ R ,
c*-expy (p) = x ⇔ p = −Dy c(x, y)
Using (2.5), we can rewrite (2.4) as
(2.6)
∂c u(x) ⊂ c-expx ∂ − u(x) .
Notice that, if c ∈ C 1 and Y is bounded, it follows immediately from (2.1) that c-convex functions
are Lipschitz, so in particular they are differentiable a.e.
The following notation will be convenient: given a c-convex function u : X → R, we define (at
almost every point) the map Tu : X → Y as
(2.7)
Tu (x) := c-expx (∇u(x)).
(Of course Tu depends also on c, but to keep the notation lighter we prefer not to make this
dependence explicit. The reader should keep in mind that, whenever we write Tu , the cost c is
always the one for which u is c-convex.)
Finally, let us observe that if c satisfies (C0) and Y is bounded, then it follows from (2.1) that u
is semiconvex (i.e., there exists a constant C > 0 such that u + C|x|2 /2 is convex, see for instance
[13]). In particular, by Alexandrov’s Theorem, c-convex functions are twice differentiable a.e. (see
[37, Theorem 14.25] for a list of different equivalent definitions of this notion).
The following is a basic result in optimal transport theory (see for instance [37, Chapter 10]):
4
G. DE PHILIPPIS AND A. FIGALLI
Theorem 2.1. Let c : X × Y → R satisfy (C0)-(C1). Given two probability densities f and
g supported on X and Y respectively, there exists a c-convex function u : X → R such that
Tu : X → Y is the unique optimal transport map sending f onto g.
In the particular case c(x, y) = −x · y (which is equivalent to the quadratic cost |x − y|2 /2),
c-convex functions are convex and the above result takes the following simple form [2]:
Theorem 2.2. Let c(x, y) = −x · y. Given two probability densities f and g supported on X and
Y respectively, there exists a convex function v : X → R such that Tv = ∇v : X → Y is the unique
optimal transport map sending f onto g.
Although on Riemannian manifolds the cost function c = d2 /2 is not smooth everywhere, one can
still prove existence of optimal maps [34, 13, 17] (let us remark that, in this case, the c-exponential
map coincides with the classical exponential map in Riemannian geometry):
Theorem 2.3. Let M be a smooth Riemannian manifold, and c = d2 /2. Given two probability
densities f and g supported on M , there exists a c-convex function u : M → R ∪ {+∞} such that
u is differentiable f -a.e., and Tu (x) = expx (∇u(x)) is the unique optimal transport map sending f
onto g.
We conclude this section by recalling that c-convex functions arising in optimal transport problems solve a Monge-Ampère type equation almost everywhere, referring to [1, Section 6.2], [37,
Chapters 11 and 12], and [15] for more details.
Whenever c satisfies (C0)-(C3), then the transport condition (Tu )] f = g gives
f (x)
a.e.
g(Tu (x))
In addition, the c-convexity of u implies that, at every point x where u is twice differentiable,
(2.9)
D2 u(x) + Dxx c x, c-expx (∇u(x)) ≥ 0.
| det(DTu (x))| =
(2.8)
Hence, writing (2.7) as
−Dx c(x, Tu (x)) = ∇u(x),
differentiating the above relation with respect to x, and using (2.8) and (2.9), we obtain
(2.10)
f (x)
det D2 u(x) + Dxx c x, c-expx (∇u(x)) = det Dxy c x, c-expx (∇u(x)) g(c-expx (∇u(x)))
at every point x where u it is twice differentiable. In particular, when c(x, y) = −x · y, the convex
function v provided by Theorem 2.2 solves the classical Monge-Ampère equation
f (x)
det D2 v(x) =
a.e.
g(∇v(x))
3. The localization argument and proof of the results
The goal of this section is to prove Theorems 1.3 and 1.4 by showing that the assumptions of
Theorems 4.3 and 5.3 below are satisfied near almost every point.
The rough idea is the following: if x̄ is a point where the semiconvex function u is twice differentiable, then around that point u looks like a parabola. In addition, by looking close enough to
x̄, the cost function c will be very close to the linear one and the densities will be almost constant
there. Hence we can apply Theorem 4.3 to deduce that u is of class C 1,β in neighborhood of x̄ (resp.
PARTIAL REGULARITY FOR OPTIMAL TRANSPORT MAPS
5
k+2,α
k,α
u is of class C k+2,α by Theorem 5.3, if c ∈ Cloc
and f, g ∈ Cloc
), which implies in particular that
k+2,α
0,β
k+1,α
Tu is of class C
in neighborhood of x̄ (resp. Tu is of class C
by Theorem 5.3, if c ∈ Cloc
k,α
). Being our assumptions completely symmetric in x and y, we can apply the same
and f, g ∈ Cloc
argument to the optimal map T ∗ sending g onto f . Since T ∗ = (Tu )−1 (see the discussion below),
0,β
it follows that Tu is a global homeomorphism of class Cloc
(resp. Tu is a global diffeomorphism of
k+1,α
class Cloc ) outside a closed set of measure zero.
We now give a detailed proof.
Proof of Theorem 1.3. Let us introduce the “c-conjugate” of u, that is, the function uc : Y → R
defined as
uc (y) := sup − c(x, y) − u(x) .
x∈X
Then
(3.1)
uc
is
c∗ -convex,
where
c∗ (y, x) := c(x, y),
and
x ∈ ∂c∗ uc (y)
⇔
y ∈ ∂c u(x)
(see for instance [37, Chapter 5]).
Being our assumptions completely symmetric in x and y, c∗ satisfies the same assumptions as c.
In particular, by Theorem 2.1, there exists an optimal map T ∗ (with respect to c∗ ) sending g onto
f . In addition, it is well-known that T ∗ is actually equal to
Tuc (y) = c*-expy ∇uc (y) ,
and that Tu and Tuc are inverse to each other, that is
(3.2)
Tuc Tu (x) = x, Tu Tuc (y) = y
for a.e. x ∈ X, y ∈ Y
(see, for instance, [1, Remark 6.2.11]).
Since semiconvex functions are twice differentiable a.e., there exist sets X1 ⊂ X, Y1 ⊂ Y of full
measure such that (3.2) holds for every x ∈ X1 and y ∈ Y1 , and in addition u is twice differentiable
for every x ∈ X1 and uc is twice differentiable for every y ∈ Y1 . Let us define
X 0 := X1 ∩ (Tu )−1 (Y1 ).
Using that Tu transports f on g and that the two densities are bounded away from zero and infinity,
we see that X 0 is of full measure in X.
We fix a point x̄ ∈ X 0 . Since u is differentiable at x̄ (being twice differentiable), it follows by
(2.6) that the set ∂c u(x̄) is a singleton, namely ∂c u(x̄) = {c-expx̄ (∇u(x̄))}. Set ȳ := c-expx̄ (∇u(x̄)).
Since ȳ ∈ Y1 (by definition of X 0 ), uc is twice differentiable at ȳ and x̄ = Tuc (ȳ). Up to a translation
in the system of coordinates (both in x and y) we can assume that both x̄ and ȳ coincide with the
origin 0.
Let us define
ū(z) := u(z) − u(0) + c(z, 0) − c(0, 0),
c̄(z, w) := c(z, w) − c(z, 0) − c(0, w) + c(0, 0),
ūc̄ (w) := uc (w) − u(0) + c(0, w) − c(0, 0).
Then ū is a c̄-convex function, ūc̄ is its c̄-conjugate, Tū = Tu , and Tūc̄ = Tuc , so in particular
(Tū )] f = g and (Tūc̄ )] g = f . In addition, because by assumption 0 ∈ X 0 , ū is twice differentiable
6
G. DE PHILIPPIS AND A. FIGALLI
at 0 and ūc̄ is twice differentiable at 0 = Tū (0). Let us define P := D2 ū(0), and M := Dxy c̄(0, 0).
Then, since c̄(·, 0) = c̄(0, ·) ≡ 0 and c̄ ∈ C 2 , a Taylor expansion gives
1
ū(z) = P z · z + o(|z|2 ),
c̄(z, w) = M z · w + o(|z|2 + |w|2 ),
2
Let us observe that, since by assumption f and g are bounded away from zero and infinity, by
(C3) and (2.10) applied to ū and c̄ we get that det(P ), det(M ) 6= 0. In addition (2.9) implies that
P is a positive definite symmetric matrix. Hence, we can perform a second change of coordinates:
z 7→ z̃ := P 1/2 z, w 7→ w̃ := −P −1/2 M ∗ w (M ∗ being the transpose of M ), so that, in the new
variables,
1
(3.3)
ũ(z̃) := ū(z) = |z̃|2 + o(|z̃|2 ),
c̃(z̃, w̃) := c̄(z, w) = −z̃ · w̃ + o(|z̃|2 + |w̃|2 ).
2
By an easy computation it follows that (Tũ )] f˜ = g̃, where 1
(3.4)
f˜(z̃) := det(P −1/2 ) f (P −1/2 z̃),
g̃(w̃) := det (M ∗ )−1 P 1/2 g(−(M ∗ )−1 P 1/2 w̃).
Notice that
(3.5)
Dz̃ z̃ c̃(0, 0) = Dw̃w̃ c̃(0, 0) = 0n×n ,
−Dz̃ w̃ c̃(0, 0) = Id,
D2 ũ(0) = Id,
so, using (2.10), we deduce that
(3.6)
det D2 ũ(0) + Dz̃ z̃ c̃(0, 0)
f˜(0)
=
= 1.
det Dz̃ w̃ c̃(0, 0) g̃(0)
To ensure that we can apply Theorems 4.3 and 5.3, we now perform the following dilation: for
ρ > 0 we define
1
1
cρ (z̃, w̃) := 2 c̄(ρz̃, ρw̃).
uρ (z̃) := 2 ū(ρz̃),
ρ
ρ
We claim that, provided ρ is sufficiently small, uρ and cρ satisfy the assumptions of Theorems 4.3
and 5.3.
Indeed, it is immediate to check that uρ is a cρ -convex function. Also, by the same argument
as above, from the relation (Tũ )] f˜ = g̃ we deduce that Tuρ sends f˜(ρz̃) onto g̃(ρw̃). In addition,
since we can freely multiply both densities by a same constant, it actually follows from (3.6) that
(Tuρ )] fρ = gρ , where
g̃(ρw̃)
f˜(ρz̃)
,
gρ (w̃) :=
.
fρ (z̃) :=
˜
g̃(0)
f (0)
In particular, since f and g are continuous, we get
(3.7)
|fρ − 1| + |gρ − 1| → 0
inside B3
as ρ → 0. Also, by (3.3) we get that, for any z̃, w̃ ∈ B3 ,
1
(3.8)
uρ (z̃) = |z̃|2 + o(1),
cρ (z̃, w̃) = −z̃ · w̃ + o(1),
2
1An easy way to check this is to observe that the measures µ := f (x)dx and ν := g(y)dy are independent of the
choice of coordinates, hence (3.4) follows from the identities
f (x)dx = f˜(x̃)dx̃,
g(y)dy = g̃(ỹ)dỹ.
PARTIAL REGULARITY FOR OPTIMAL TRANSPORT MAPS
7
where o(1) → 0 as ρ → 0. In particular, (4.9) and (4.10) hold with any positive constants δ0 , η0
provided ρ is small enough.
Furthermore, by the second order differentiability of ũ at 0 it follows that the multivalued map
z̃ 7→ ∂ − ū(z̃) is differentiable at 0 (see [37, Theorem 14.25]) with gradient equal to the identity
matrix (see (3.3)), hence
∂ − uρ (z̃) ⊂ Bγρ (z̃)
∀ z̃ ∈ B2 ,
where γρ → 0 as ρ → 0. Since ∂cρ uρ ⊂ cρ -exp (∂ − uρ ) (by (2.6)) and k cρ -exp − Id k∞ = o(1) (by
(3.8)), we get
∂cρ uρ (z̃) ⊂ Bδρ (z̃)
(3.9)
∀ z̃ ∈ B3 ,
with δρ = o(1) as ρ → 0. Moreover, the cρ -conjugate of uρ is easily seen to be
c
uρρ (w̃) = ūc̄ ρ(M ∗ )−1 P 1/2 w̃ .
c
Since uc is twice differentiable at 0, so is uρρ . In addition, an easy computation
c
D2 uρρ (0) = Id. Hence, arguing as above we obtain that
c
∂c∗ρ uρρ (w̃) ⊂ Bδρ0 (w̃)
(3.10)
2
shows that
∀ w̃ ∈ B3 ,
with δρ0 = o(1) as ρ → 0.
We now define
C1 := B 1 ,
C2 := ∂cρ uρ (C1 ).
Observe that both C1 and C2 are closed (since the c-subdifferential of a compact set is closed). Also,
thanks to (3.9), by choosing ρ small enough we can ensure that B1/3 ⊂ C2 ⊂ B3 . Finally, it follows
from (2.6) that
c (Tuρ )−1 (C2 ) \ C1 ⊂ (Tuρ )−1 {points of non-differentiability of uρρ } ,
and since this latter set has measure zero, a simple computation shows that
Tuρ ] (fρ 1C1 ) = gρ 1C2 .
Thus, thanks to (4.8), we get that for any β < 1 the assumptions of Theorem 4.3 are satisfied,
k+2,α
k,α
provided we choose ρ sufficiently small. Moreover, if in addition c ∈ Cloc
(X × Y ), f ∈ Cloc
(X),
k,α
and g ∈ Cloc (Y ), then also the assumptions of Theorem 5.3 are satisfied.
Hence, by applying Theorem 4.3 (resp. Theorem 5.3) we deduce that uρ ∈ C 1,β (B1/7 ) (resp.
uρ ∈ C k+2,α (B1/9 )), so going back to the original variables we get the existence of a neighborhood Ux̄
of x̄ such that u ∈ C 1,β (Ux̄ ) (resp. u ∈ C k+2,α (Ux̄ )). This implies in particular that Tu ∈ C 0,β (Ux̄ )
(resp. Tu ∈ C k+1,α (Ux̄ )). Moreover, it follows by Corollary 4.6 that Tu (Ux̄ ) contains a neighborhood
of ȳ.
c
We now observe that, by symmetry, we can also apply Theorem 4.3 (resp. Theorem 5.3) to uρρ .
Hence, there exists a neighborhood Vȳ of ȳ such that Tuc ∈ C 0,β (Vȳ ). Since Tu and Tuc are inverse
2For instance, this follows by differentiating both relations
Dz̃ cρ z̃, Tuρ (z̃) = −∇uρ (z̃)
and
c
Dw̃ cρ Tucρρ (w̃), w̃ = −∇uρρ (w̃)
at 0, and using then (3.5) and the fact that ∇Tucρρ (0) = [∇Tuρ (0)]−1 and D2 uρ (0) = Id.
8
G. DE PHILIPPIS AND A. FIGALLI
to each other (see (3.2)) we deduce that, possibly reducing the size of Ux̄ , Tu is a homeomorphism
(resp. diffeomorphism) between Ux̄ and Tu (Ux̄ ). Let us consider the open sets
[
[
X 00 :=
Ux̄ ,
Y 00 :=
Tu (Ux̄ ),
x̄∈X 0
x̄∈X 0
and define the (relatively) closed ΣX := X \ X 00 , ΣY := Y \ Y 00 . Since X 00 ⊃ X 0 , X 00 is a set of full
measure, so |ΣX | = 0. In addition, since ΣY = Y \ Y 00 ⊂ Y \ Tu (X 0 ) and Tu (X 0 ) has full measure
in Y , we also get that |ΣY | = 0.
Finally, since Tu : X \ ΣX → Y \ ΣY is a local homeomorphism (resp. diffeomorphism), by (3.2)
it follows that Tu : X \ ΣX → Y \ ΣY is a global homeomorphism (resp. diffeomorphism), which
concludes the proof.
Proof of Theorem 1.4. The only difference with respect to the situation in Theorem 1.3 is that now
the cost function c = d2 /2 is not smooth on the whole M × M . However, even if d2 /2 is not
everywhere smooth and M is not necessarily compact, it is still true that the c-convex function
u provided by Theorem 2.3 is locally semiconvex (i.e., it is locally semiconvex when seen in any
chart) [13, 17]. In addition, as shown in [9, Proposition 4.1] (see also [14, Section 3]), if u is
twice differentiable at x, then the point Tu (x) is not in the cut-locus of x. Since the cut-locus is
closed and d2 /2 is smooth outside the cut-locus, we deduce the existence of a set X of full measure
such that, if x0 ∈ X, then: (1) u is twice differentiable at x0 ; (2) there exists a neighborhood
Ux0 × VTu (x0 ) ⊂ M × M of (x0 , Tu (x0 )) such that c ∈ C ∞ (Ux0 × VTu (x0 ) ). Hence, by taking
a local chart around (x0 , Tu (x0 )), the same proof as the one of Theorem 1.3 shows that Tu is
a local homeomorphism (resp. diffeomorphism) around almost every point. Using as before that
Tu : M → M is invertible a.e., it follows that Tu is a global homeomorphism (resp. diffeomorphism)
outside a closed singular set of measure zero. We leave the details to the interested reader.
4. C 1,β regularity and strict c-convexity
In this and the next section we prove that, if in some open set a c-convex function u is sufficiently
close to a parabola and the cost function is close to the linear one, then u is smooth in some smaller
set.
The idea of the proof (which is reminiscent of the argument introduced by Caffarelli in [6] to
show W 2,p and C 2,α estimates for the classical Monge-Ampère equation, though several additional
complications arise in our case) is the following: since the cost function is close to the linear one and
both densities are almost constant, u is close to a convex function v solving an optimal transport
problem with linear cost and constant densities (Lemma 4.1). In addition, since u is close to a
parabola, so is v. Hence, by [18] and Caffarelli’s regularity theory, v is smooth, and we can use this
information to deduce that u is even closer to a second parabola (given by the second order Taylor
expansion of v at the origin) inside a small neighborhood around of origin. By rescaling back this
neighborhood at scale 1 and iterating this construction, we obtain that u is C 1,β at the origin for
some β ∈ (0, 1). Since this argument can be applied at every point in a neighborhood of the origin,
we deduce that u is C 1,β there, see Theorem 4.3. (A similar strategy has also been used in [7] to
show regularity optimal transport maps for the cost |x − y|p , either when p is close to 2 or when X
and Y are sufficiently far from each other.)
Once this result is proved, we know that ∂ − u is a singleton at every point, so it follows from
(2.6) that
∂c u(x) = c-expx (∂ − u(x)),
PARTIAL REGULARITY FOR OPTIMAL TRANSPORT MAPS
9
see Remark 4.4 below. (The above identity is exactly what in general may fail for general c-convex
functions, unless the MTW condition holds [31].) Thanks to this fact, we obtain that u enjoys a
comparison principle (Proposition 5.2), and this allows us to use a second approximation argument
with solutions of the classical Monge-Ampère equation (in the spirit of [6, 27]) to conclude that u
0
is C 2,σ in a smaller neighborhood, for some σ 0 > 0. Then higher regularity follows from standard
elliptic estimates, see Theorem 5.3.
Lemma 4.1. Let C1 and C2 be two closed sets such that
B1/K ⊂ C1 , C2 ⊂ BK
(4.1)
for some K ≥ 1, f and g two densities supported respectively in C1 and C2 , and u : C1 → R a
c-convex function such that ∂c u(C1 ) ⊂ BK and (Tu )] f = g. Let ρ > 0 be such that |C1 | = |ρ C2 |
(where ρ C2 denotes the dilation of C2 with respect to the origin), and let v be a convex function
such that ∇v] 1C1 = 1ρC2 and v(0) = u(0). Then there exists an increasing function ω : R+ → R+ ,
depending only K, and satisfying ω(δ) ≥ δ and ω(0+ ) = 0, such that, if
kf − 1C1 k∞ + kg − 1C2 k∞ ≤ δ
(4.2)
and
kc(x, y) + x · ykC 2 (BK ×BK ) ≤ δ,
(4.3)
then
ku − vkC 0 (B1/K ) ≤ ω(δ).
Proof. Assume the lemma is false. Then there exists ε0 > 0, a sequence of closed sets C1h , C2h
satisfying (4.1), functions fh , gh satisfying (4.2) with δ = 1/h, and costs ch converging in C 2 to
−x · y, such that
uh (0) = vh (0) = 0 and kuh − vh kC 0 (B1/K ) ≥ ε0 ,
where uh and vh are as in the statement. First, we extend uh an vh to BK as
uh (z)−ch (x, y)+ch (z, y) ,
vh (x) :=
sup
vh (z)+p·(x−z) .
uh (x) :=
sup
z∈C1h , p∈∂ − vh (z)
z∈C1h , y∈∂ch uh (z)
h
Notice that,
R since Rby assumption ∂ch uh (C1 ) ⊂ BK , we have ∂ch uh (BK ) ⊂ BK . Also, (Tuh )] fh = gh
gives that fh = gh , so it follows from (4.2) that
1/n
ρh = |C1h |/|C2h |
→1
as h → ∞,
which implies that ∂ − vh (BK ) ⊂ Bρh K ⊂ B2K for h large. Thus, since the C 1 -norm of ch is uniformly
bounded, we deduce that both uh and vh are uniformly Lipschitz. Recalling that uh (0) = vh (0) =
0, we get that, up to a subsequence, uh and vh uniformly converge inside BK to u∞ and v∞
respectively, where
(4.4)
u∞ (0) = v∞ (0) = 0
and
ku∞ − v∞ kC 0 (B1/K ) ≥ ε0 .
In addition fh (resp. gh ) weak-∗ converge in L∞ to some density f∞ (resp. g∞ ) supported in B K .
Also, since ρh → 1, using (4.2) we get that 1C h (resp. 1ρh C h ) weak-∗ converges in L∞ to f∞ (resp.
1
2
g∞ ). Finally we remark that, because of (4.2) and the fact that C1h ⊃ B1/K , we also have
f∞ ≥ 1B1/K .
10
G. DE PHILIPPIS AND A. FIGALLI
In order to get a contradiction we have to show that u∞ = v∞ in B1/K . To see this, we apply
[37, Theorem 5.20] to deduce that both ∇u∞ and ∇v∞ are optimal transport maps for the linear
cost −x · y sending f∞ onto g∞ . By uniqueness of the optimal map (see Theorem 2.2) we deduce
that ∇v∞ = ∇u∞ almost everywhere inside B1/K ⊂ spt f∞ , hence u∞ = v∞ in B1/K (since
u∞ (0) = v∞ (0) = 0), contradicting (4.4).
Here and in the sequel, we use Nr (E) to denote the r-neighborhood of a set E.
Lemma 4.2. Let u and v be, respectively, c-convex and convex, let D ∈ Rn×n be a symmetric
matrix satisfying
Id /K ≤ D ≤ K Id
(4.5)
for some K ≥ 1, and define the ellipsoid
E(x0 , h) := x : D(x − x0 ) · (x − x0 ) ≤ h ,
h > 0.
Assume that there exist small positive constants ε, δ such that
(4.6)
kv − ukC 0 (E(x0 ,h)) ≤ ε,
kc + x · ykC 2 (E(x0 ,h)×∂c u(E(x0 ,h)) ≤ δ.
Then
(4.7)
∂c u E(x0 , h −
√ ε) ⊂ NK 0 (δ+√hε) ∂v(E(x0 , h))
∀ 0 < ε < h2 ≤ 1,
where K 0 depends only on K.
Proof. Up to a change of coordinates we can assume that x0 = 0, and to simplify notation we set
Eh := E(x0 , h). Let us define
√
v̄(x) := v(x) + ε + 2 ε(Dx · x − h),
so that v̄ ≥ u outside Eh , and v̄ ≤ u inside Eh−√ε . Then, taking a c-support to u in Eh−√ε (i.e., a
function Cx,y as in (2.3), with x ∈ Eh−√ε and y ∈ ∂c u(x)), moving it down and then lifting it up
until it touches v̄ from below, we see that it has to touch the graph of v̄ at some point x̄ ∈ Eh : in
other words 3
∂c u(Eh−√ε ) ⊂ ∂c v̄(Eh ).
√
By (4.5) we see that diam Eh ≤ 2 Kh, so by a simple computation (using again (4.5)) we get
∂ − v̄(Eh ) ⊂ N4K √Khε ∂ − v(Eh ) .
Thus, since ∂c v̄(Eh ) ⊂ c-exp ∂ − v̄(Eh ) (by (2.6)) and k c-exp − Id kC 0 ≤ δ (by (4.6)), we easily
deduce that
∂c u(Eh−√ε ) ⊂ NK 0 (δ+√hε) ∂ − v(Eh ) ,
proving (4.7).
Theorem 4.3. Let C1 and C2 be two closed sets satisfying
B1/3 ⊂ C1 , C2 ⊂ B3 ,
3Even if v̄ is not c-convex, it still makes sense to consider his c-subdifferential (notice that the c-subdifferential of
v̄ may be empty at some points). In particular, the inclusion ∂c v̄(x) ⊂ c-expx (∂ − v̄(x)) still holds.
PARTIAL REGULARITY FOR OPTIMAL TRANSPORT MAPS
11
let f, g be two densities supported in C1 and C2 respectively, and let u : C1 → R be a c-convex
function such that ∂c u(C1 ) ⊂ B3 and (Tu )] f = g. Then, for every β ∈ (0, 1) there exist constants
δ0 , η0 > 0 such that the following holds: if
(4.8)
kf − 1C1 k∞ + kg − 1C2 k∞ ≤ δ0 ,
(4.9)
kc(x, y) + x · ykC 2 (B3 ×B3 ) ≤ δ0 ,
and
1
2
u − |x| ≤ η0 ,
0
2
C (B3 )
(4.10)
then u ∈ C 1,β (B1/7 ).
Proof. We divide the proof into several steps.
• Step 1: u is close to a strictly convex solution of the Monge Ampère equation. Let v : Rn → R
be a convex function such that ∇v] 1C1 = 1ρC2 with ρ = (|C1 |/|C2 |)1/n (see Theorem 2.2). Up to
adding a constant to v, without loss of generality we can assume that v(0) = u(0). Hence, we can
apply Lemma 4.1 to obtain
kv − ukC 0 (B1/3 ) ≤ ω(δ0 )
(4.11)
for some (universal) modulus of continuity ω : R+ → R+ , which combined with (4.10) gives
v − 1 |x|2 ≤ η0 + ω(δ0 ).
0
2
C (B1/3 )
R
R
Also, since C1 f = C2 g, it follows easily from (4.8) that |ρ − 1| ≤ 3δ0 . By these two facts we
get that ∂ − v(B1/4 ) ⊂ B7/24 ⊂ ρC2 provided δ0 and η0 are small enough (recall that v is convex
and that B1/3 ⊂ C2 ), so we can apply [18, Proposition 3.4] to deduce that v is a strictly convex
Alexandrov solution to the Monge-Ampère equation
det D2 v = 1
(4.12)
in B1/4 .
In addition, by a simple compactness argument, we see that the modulus of strict convexity of v
inside B1/4 is universal. So, by classical Pogorelov and Schauder estimates, we obtain the existence
of a universal constant K0 ≥ 1 such that
(4.13)
kvkC 3 (B1/5 ) ≤ K0 ,
Id /K0 ≤ D2 v ≤ K0 Id
in B1/5 .
In particular, there exists a universal value h̄ > 0 such that, for all x ∈ B1/7 ,
Q(x, v, h) := z : v(z) ≤ v(x) + ∇v(x) · (z − x) + h ⊂⊂ B1/6 ∀ h ≤ h̄.
• Step 2: Sections of u are close to sections of v. Given x ∈ B1/7 and y ∈ ∂c u(x), we define
S(x, y, u, h) := z : u(z) ≤ u(x) − c(z, y) + c(x, y) + h .
We claim that, if δ0 is small enough, then for all x ∈ B1/7 , y ∈ ∂c u(x), and h ≤ h̄/2, it holds
p
p
(4.14)
Q(x, v, h − K1 ω(δ0 )) ⊂ S(x, y, u, h) ⊂ Q(x, v, h + K1 ω(δ0 )),
where K1 > 0 is a universal constant.
12
G. DE PHILIPPIS AND A. FIGALLI
Let us show the first inclusion. For this, take x ∈ B1/7 , y ∈ ∂c u(x), and define
px := −Dx c(x, y) ∈ ∂ − u(x).
Since v has universal C 2 bounds (see (4.13)) and u is semi-convex (with a universal bound), a
simple interpolation argument gives
q
p
(4.15)
|px − ∇v(x)| ≤ K 0 ku − vkC 0 (B1/5 ) ≤ K 0 ω(δ0 ) ∀ x ∈ B1/7 .
In addition, by (4.9),
|y − px | ≤ kDx c + IdkC 0 (B3 ×B3 ) ≤ δ0 ,
(4.16)
hence
|z · px + c(z, y)| ≤ |z · px − z · y| + |z · y + c(z, y)| ≤ 2δ0
∀ x, z ∈ B1/7 .
p
Thus, if z ∈ Q(x, v, h − K1 ω(δ0 )), by (4.11), (4.15), and (4.17) we get
p
u(z) ≤ v(z) + ω(δ0 ) ≤ v(x) + ∇v(x) · (z − x) + h − K1 ω(δ0 ) + ω(δ0 )
p
p
≤ u(x) + px · z − px · x + h − K1 ω(δ0 ) + 2ω(δ0 ) + 2K 0 ω(δ0 )
p
p
≤ u(x) − c(z, y) + c(x, y) + h − K1 ω(δ0 ) + 2ω(δ0 ) + 2K 0 ω(δ0 ) + 4δ0
(4.17)
≤ u(x) − c(z, y) + c(x, y) + h,
provided K1 > 0 is sufficiently large. This proves the first inclusion, and the second is analogous.
• Step 3: Both the sections of u and their images are close to ellipsoids with controlled eccentricity,
and u is close to a smooth function near x0 . We claim that there exists a universal constant
K2 ≥ 1 such that the following holds: For every η0 > 0 small, there exist small positive constants
h0 = h0 (η0 ) and δ0 = δ0 (h0 , η0 ) such that, for all x0 ∈ B1/7 , there is a symmetric matrix A satisfying
Id /K2 ≤ A ≤ K2 Id,
(4.18)
det(A) = 1,
and such that, for all y0 ∈ ∂c u(x0 ),
A B√h /8 (x0 ) ⊂ S(x0 , y0 , u, h0 ) ⊂ A B√8h0 (x0 ) ,
0
(4.19)
−1
A
B√h /8 (y0 ) ⊂ ∂c u(S(x0 , y0 , u, h0 )) ⊂ A−1 B√8h0 (y0 ) .
0
Moreover
(4.20)
u − Cx ,y − 1 A−1 (x − x0 )2 0 0
0
2
C A B√
8h0
(x0 )
≤ η0 h0 ,
where Cx0 y0 is a c-support function for u at x0 , see (2.3).
p
In order to prove the claim, take h0 h̄ small (to be fixed) and δ0 h0 such that K1 ω(δ0 ) ≤
h0 /2, where K1 is as in Step 2, so that
(4.21)
Q(x0 , v, h0 /2) ⊂ S(x0 , y0 , u, h0 ) ⊂ Q(x0 , v, 3h0 /2) ⊂⊂ B1/6 .
By (4.13) and Taylor formula we get
(4.22)
1
v(x) = v(x0 ) + ∇v(x0 ) · (x − x0 ) + D2 v(x0 )(x − x0 ) · (x − x0 ) + O(|x − x0 |3 ),
2
PARTIAL REGULARITY FOR OPTIMAL TRANSPORT MAPS
13
so that defining
(
(4.23)
1
x : D2 v(x0 )(x − x0 ) · (x − x0 ) ≤ h0
2
E(x0 , h0 ) :=
)
and using (4.13), we deduce that, for every h0 universally small,
E(x0 , h0 /2) ⊂ Q(x0 , v, h0 ) ⊂ E(x0 , 2h0 ).
(4.24)
Moreover, always for h0 small, thanks to (4.22) and the uniform convexity of v
(4.25)
∇v E(x0 , h0 ) ⊂ E ∗ (∇v(x0 ), 2h0 ) ⊂ ∇v E(x0 , 3h0 ))
where we have set
)
−1
1 2
D v(ȳ) (y − ȳ) · (y − ȳ) ≤ h0 .
y:
2
(
∗
E (ȳ, h0 ) :=
By Lemma 4.2, (4.24), and (4.25) applied with 3h0 in place of h0 , we deduce that for δ0 h0 h̄
(4.26)
∂c u S(x0 , y0 , u, h0 ) ⊂ NK 00 √ω(δ ) ∇v(E(x0 , 3h0 )) ⊂ E ∗ (∇v(x0 ), 7h0 ).
0
Moreover, by (4.15), if y0 ∈ ∂c u(x0 ) and we set px0 := −Dx c(x0 , y0 ), then
p
|y0 − ∇v(x0 )| ≤ |px0 − ∇v(x0 )| + kDx c + IdkC 0 (B3 ×B3 ) ≤ K 0 ω(δ0 ) + δ0 .
Thus, choosing δ0 sufficiently small, it holds
(4.27)
E ∗ (∇v(x0 ), 7h0 ) ⊂ E ∗ (y0 , 8h0 )
∀ y0 ∈ ∂c u(x0 ).
We now want to show that
E ∗ (y0 , h0 /8) ⊂ ∂c u S(x0 , y0 , u, h0 )
∀ y0 ∈ ∂c u(x0 ).
Observe that, arguing as above, we get
(4.28)
E ∗ (y0 , h0 /8) ⊂ E ∗ (∇v(x0 ), h0 /7) ∀ y0 ∈ ∂c u(x0 )
provided δ0 is small enough, so it is enough to prove that
E ∗ (∇v(x0 ), h0 /7) ⊂ ∂c u S(x0 , y0 , u, h0 ) .
For this, let us define the c∗ -convex function uc : B3 → R and the convex function v ∗ : B3 → R as
uc (y) := sup
− c(x, y) − u(x) ,
v ∗ (y) := sup x · y − v(x)
x∈B1/5
x∈B1/5
(see (3.1)). Then it is immediate to check that
|uc − v ∗ | ≤ ω(δ0 ) + δ0 ≤ 2ω(δ0 )
(4.29)
Also, in view of (4.13),
addition, since
(4.30)
v∗
on B3 .
is a uniformly convex function of class C 3 on the open set ∇v(B1/5 ). In
F ⊂ ∂c u(∂c∗ uc (F ))
for any set F ,
thanks to (4.21) and (4.24) it is enough to show
(4.31)
∂c∗ uc (E ∗ (∇v(x0 ), h0 /7)) ⊂ E(x0 , h0 /4).
For this, we apply Lemma 4.2 to uc and v ∗ to infer
∂c∗ uc (E ∗ (∇v(x0 ), h0 /7)) ⊂ NK 000 √ω(δ) ∇v ∗ (E ∗ (∇v(x0 ), h0 /7)) ⊂ E(x0 , h0 /4),
14
G. DE PHILIPPIS AND A. FIGALLI
where we used that
∇v ∗ = [∇v]−1
and D2 v ∗ (∇v(x0 )) = [D2 v(x0 )]−1 .
Thus, recalling (4.26), we have proved that there exist h0 universally small, and δ0 small depending
on h0 , such that
(4.32)
E ∗ (∇v(x0 ), h0 /7) ⊂ ∂c u(S(x0 , y0 , u, h0 )) ⊂ E ∗ (∇v(x0 ), 7h0 )
∀ x0 ∈ B1/7 .
Using (4.21), (4.24), (4.27), and (4.28), this proves (4.19) with A := [D2 v(x0 )]−1/2 . Also, thanks
to (4.12) and (4.13), (4.18) holds.
In order to prove the second part of the claim, we exploit (4.11), (4.9), (4.16), (4.15), (4.22), and
(4.18) (recall that Cx0 ,y0 is defined in (2.3) and that A = [D2 v(x0 )]−1/2 ):
−1
2 1
u − Cx ,y − A (x − x0 ) 0 0
0
2
C (E(x0 ,8h0 ))
1 2
= u − Cx0 ,y0 − D v(x0 )(x − x0 ) · (x − x0 )
0
2
C (E(x0 ,8h0 ))
≤ 2ku − vkC 0 (E(x0 ,8h0 )) + kc(x, y0 ) + x · y0 kC 0 (E(x0 ,8h0 )) + kc(x0 , y0 ) + x0 · y0 kC 0 (E(x0 ,8h0 ))
+ y0 − px0 · (x − x0 )C 0 (E(x0 ,8h0 )) + px0 − ∇v(x0 ) · (x − x0 )C 0 (E(x0 ,8h0 ))
1
2
+
v − v(x0 ) − ∇v(x0 ) · (x − x0 ) − 2 D v(x0 )(x − x0 ) · (x − x0 ) 0
C (E(x0 ,8h0 ))
p
3
p
≤ 2ω(δ0 ) + 3δ0 + K 0 ω(δ0 ) + K K2 8h0 ≤ η0 h0 ,
where the last inequality follows by choosing first h0 sufficiently small, and then δ0 much smaller
than h0 .
• Step 4: A first change of variables. Fix x0 ∈ B1/7 , y0 ∈ ∂c u(x0 ), define M := −Dxy c(x0 , y0 ), and
consider the change of variables
(
x̄ := x − x0
ȳ := M −1 (y − y0 ).
Notice that, by (4.9), it follows that
(4.33)
|M − Id | + |M −1 − Id | ≤ 3δ0
for δ0 sufficiently small. We also define
c̄(x̄, ȳ) := c(x, y) − c(x, y0 ) − c(x0 , y) + c(x0 , y0 ),
ū(x̄) := u(x) − u(x0 ) + c(x, y0 ) − c(x0 , y0 ),
ūc̄ (ȳ) := uc (y) − uc (y0 ) + c(x0 , y) − c(x0 , y0 ).
Then ū is c̄-convex, ūc̄ is c̄∗ -convex (where c̄∗ (ȳ, x̄) = c̄(x̄, ȳ)), and
(4.34)
c̄(·, 0) = c̄(0, ·) ≡ 0,
Dx̄ȳ c̄(0, 0) = − Id .
We also notice that
(4.35)
∂c̄ ū(x̄) = M −1 ∂c u(x̄ + x0 ) − y0 .
PARTIAL REGULARITY FOR OPTIMAL TRANSPORT MAPS
15
Thus, recalling (4.19), and using (4.33) and (4.35), for δ0 sufficiently small we obtain
(4.36)
A B√h /9 ⊂ S(0, 0, ū, h0 ) ⊂ A B√9h0 ,
0
−1
√
A
B h /9 ⊂ M −1 A−1 B√h /8 ⊂ ∂c̄ ū(S(0, 0, ū, h0 )) ⊂ M −1 A−1 B√8h0 ⊂ A−1 B√9h0 .
0
0
Since (Tu )] f = g, it follows that Tū = c̄-exp(∇ū) satisfies
(Tū )] f¯ = ḡ,
with f¯(x̄) := f (x̄ + x0 ), ḡ(ȳ) := det(M ) g(M ȳ + y0 )
(see for instance the footnote in the proof of Theorem 1.3). Notice that, since |M − Id | ≤ δ0 (by
(4.9)), we have | det(M ) − 1| ≤ (1 + 2n)δ0 (for δ0 small), so by (4.8) we get
(4.37)
kf¯ − 1C −x k∞ + kḡ − 1M −1 (C −y ) k∞ ≤ 2(1 + n)δ0 .
1
0
2
0
• Step 5: A second change of variables and the iteration argument. We now perform a second
change of variable: we set
(
x̃ := √1h A−1 x̄
0
(4.38)
ỹ := √1h Aȳ,
0
and define
p
1 p
c̄ h0 Ax̃, h0 A−1 ỹ ,
h0
1 p
u1 (x̃) :=
ū( h0 Ax̃),
h0
1 c̄ p
u1c1 (ỹ) :=
ū ( h0 A−1 ỹ).
h0
We also define
p
p
g1 (ỹ) := ḡ( h0 A−1 ỹ).
f1 (x̃) := f¯( h0 Ax̃),
Since det(A) = 1 (see (4.18)), it is easy to check
√ that (Tu1 )] f1 = g1 (see the footnote in the proof
of Theorem 1.3). Also, since kAk + kA−1 k h0 1, it follows from (4.37) that
c1 (x̃, ỹ) :=
|f1 − 1| + |g1 − 1| ≤ 2(1 + n)δ0
(4.39)
inside B3 .
Moreover, defining
(1)
(1)
C1 := S(0, 0, u1 , 1),
(1)
both C1
(1)
and C2
C2 := ∂c1 u1 (S(0, 0, u1 , 1)),
are closed, and thanks to (4.36)
(1)
(1)
B1/3 ⊂ C1 , C2 ⊂ B3 .
(4.40)
Also, since (Tu1 )] f1 = g1 , arguing as in the proof of Theorem 1.3 we get
(Tu1 )] f1 1C (1) = g1 1C (1) ,
1
2
and by (4.39)
kf1 1C (1) − 1C (1) k∞ + kg1 1C (1) − 1C (1) k∞ ≤ 2(1 + n)δ0 .
1
1
2
2
Finally, by (4.34) and (4.20), it is easy to check that
kc1 (x̃, ỹ) + x̃ · ỹkC 2 (B3 ×B3 ) ≤ δ0 ,
u1 − 1 |x̃|2 ≤ η0 .
0
2
C (B3 )
16
G. DE PHILIPPIS AND A. FIGALLI
This shows that u1 satisfies the same assumptions as u with δ0 replaced by 2(1 + n)δ0 . Hence, up
to take δ0 slightly smaller, we can apply Step 3 to u1 , and we find a symmetric matrix A1 satisfying
Id /K2 ≤ A1 ≤ K2 Id,
det(A1 ) = 1,
A1 B√h /8 ⊂ S(0, 0, u1 , h0 ) ⊂ A1 B√8h0 ,
0
B√8h0 ,
A−1
B√h /8 ⊂ ∂c1 u1 (S(0, 0, u1 , h0 )) ⊂ A−1
1
1
0
u1 − 1 A−1 x̃2 ≤ η0 h0 .
0
√
2 1
C (A1 (B(0, 8h0 ))
(Here K2 and h0 are as in Step 3.)
This allows us to apply to u1 the very same construction as the one used above to define u1 from
ū: we set
p
p
p
1
1
c2 (x̃, ỹ) :=
u2 (x̃) :=
h0 A1 x̃, h0 A−1
c1
u1 ( h0 A1 x̃),
1 ỹ ,
h0
h0
so that (Tu2 )] f2 = g2 with
p
p
g2 (ỹ) := ḡ( h0 A−1
f2 (x̃) := f1 ( h0 A1 x̃),
1 ỹ).
Arguing as before, it is easy to check that u2 , c2 , f2 , g2 satisfy the same assumptions as u1 , c1 , f1 , g1
with exactly the same constants.
So we can keep iterating this construction, defining for any k ∈ N
p
p
p
1
1
ỹ ,
uk+1 (x̃) :=
ck+1 (x̃, ỹ) :=
ck
h0 Ak x̃, h0 A−1
uk ( h0 Ak x̃),
k
h0
h0
where Ak is the matrix constructed in the k-th iteration. In this way, if we set
Mk := Ak · . . . · A1 ,
∀ k ≥ 1,
we obtain a sequence of symmetric matrices satisfying
Id /K2k ≤ Mk ≤ K2k Id,
(4.41)
det(Mk ) = 1,
and such that
Mk B(h0 /8)k/2 ⊂ S(0, 0, uk , hk0 ) ⊂ Mk B(8h0 )k/2 .
(4.42)
• Step 6: C 1,β regularity. We now show that, for any β ∈ (0, 1), we can choose h0 and δ0 = δ0 (h0 )
small enough so that u1 is C 1,β at the origin (here u1 is the function constructed in the previous
step). This will imply that u is C 1,β at x0 with universal bounds, which by the arbitrariness of
x0 ∈ B1/7 gives u ∈ C 1,β (B1/7 ).
Fix β ∈ (0, 1). Then by (4.41) and (4.42) we get
(4.43)
so defining r0 :=
B
√
√
k
√
h0 /( 8K2 )
⊂ S(0, 0, u1 , hk0 ) ⊂ B
k ,
√
K2 8h0
√
h0 /( 8K2 ) we obtain
ku1 kC 0 (B
k
r0
)
≤ hk0 =
√
8K2 r0
2k
(1+β)k
≤ r0
,
provided h0 (and so r0 ) is sufficiently small. This implies the C 1,β regularity of u1 at 0, concluding
the proof.
PARTIAL REGULARITY FOR OPTIMAL TRANSPORT MAPS
17
Remark 4.4 (Local to global principle). If u is differentiable at x and c satisfies (C0)-(C1), then
every “local support” at x is also a “global c-support” at x, that is, ∂c u(x) = c-expx (∂ − u(x)). To
see this, just notice that
∅=
6 ∂c u(x) ⊂ c-expx (∂ − u(x)) = {c-expx (∇u(x))}
(recall (2.6)), so necessarily the two sets have to coincide.
Corollary 4.5. Let u be as in Theorem 4.3. Then u is strictly c-convex in B1/7 . More precisely,
for every γ > 2 there exist η0 , δ0 > 0 depending only on γ such that, if the hypotheses of Theorem
4.3 are satisfied, then, for all x0 ∈ B1/7 , y0 ∈ ∂c u(x0 ), and Cx0 ,y0 as in (2.3), we have
(4.44)
inf
∀ r ≤ dist(x0 , ∂B1/7 ),
u − Cx0 ,y0 ≥ c0 rγ
∂Br (x0 )
with c0 > 0 universal.
Proof. With the same notation as in the proof of Theorem 4.3, it is enough to show that
inf u1 ≥ r1/β ,
∂Br
√
where u1 is the function constructed in Step 5 of the proof of Theorem 4.3. Defining %0 := K2 8h0 ,
it follows from (4.43) that
2k
√
inf u1 ≥ hk0 = %0 /( 8K2 )
≥ %γk
0 ,
∂B%k
0
provided h0 is small enough.
A simple consequence of the above results is the following:
Corollary 4.6. Let u be as in Theorem 4.3, then Tu (B1/7 ) is open.
Proof. Since u ∈ C 1,β (B1/7 ) we have that Tu (B1/7 ) = ∂c u(B1/7 ) (see Remark 4.4). We claim that
it is enough to show that if y0 ∈ ∂c u(B1/7 ), then there exists ε = ε(y0 ) > 0 small such that, for all
|y − y0 | < ε, the function u(·) + c(·, y) has a local minimum at some point x̄ ∈ B1/7 . Indeed, if this
is the case, then
∇u(x̄) = −Dx c(x̄, y),
and so y ∈ ∂c u(x̄) (by Remark 4.4), hence Bε (y0 ) ⊂ Tu (B1/7 ).
To prove the above fact, fix r > 0 such that Br (x0 ) ⊂ B1/7 , and pick x̄ a point in B r (x0 ) where
the function u(·) + c(·, y) attains its minimum, i.e.,
x̄ ∈ argmin u(x) + c(x, y) .
B r (x0 )
Since, by (4.44),
min
x∈∂Br (x0 )
u(x) + c(x, y) ≥
min
x∈∂Br (x0 )
u(x) + c(x, y0 ) − εkckC 1
≥ u(x0 ) + c(x0 , y0 ) + c0 rγ − εkckC 1 ,
while
u(x0 ) + c(x0 , y) ≤ c(x0 , y0 ) + u(x0 ) + εkckC 1 ,
choosing ε <
we obtain that x̄ ∈ Br (x0 ) ⊂ B1/7 . This implies that x̄ is a local minimum
for u(·) + c(·, y), concluding the proof.
c0
γ
2kckC 1 r
18
G. DE PHILIPPIS AND A. FIGALLI
5. Comparison principle and C 2,α regularity
We begin this section with a change of variable formula for the c-exponential map.
Lemma 5.1. Let Ω be an open set, v ∈ C 2 (Ω), and assume that ∇v(Ω) ⊂ Dom c-exp and that
D2 v(x) + Dxx c x, c-expx (∇v(x)) ≥ 0 ∀ x ∈ Ω.
Then, for every Borel set A ⊂ Ω,
Z
| c-exp(∇v(A))| ≤
A
det D2 v(x) + Dxx c x, c-expx (∇v(x))
dx.
det Dxy c x, c-expx (∇v(x)) In addition, if the map x 7→ c-expx (∇v(x)) is injective, then equality holds.
Proof. The result follows from a direct application of the Area Formula [12, Section 3.3.2, Theorem
1] once one notices that, differentiating the identity
∇v(x) = −Dx c x, c-expx (∇v(x))
(see (2.5)), the Jacobian determinant of the C 1 map x 7→ c-expx (∇v(x)) is given precisely by
det D2 v(x) + Dxx c x, c-expx (∇v(x))
.
det Dxy c x, c-expx (∇v(x)) In the next proposition we show a comparison principle between C 1 c-convex functions and
smooth solutions to the Monge-Ampère equation. 4 As already mentioned at the beginning of
Section 4 (see also Remark 4.4), the C 1 regularity of u is crucial to ensure that the c-subdifferential
coincides with its local counterpart c-exp(∂ − u).
Here and in the sequel, we use co[E] to denote the convex hull of a set E. Also, recall that Nr (E)
denotes the r-neighborhood of E.
Proposition 5.2 (Comparison principle). Let u be a c-convex function of class C 1 inside the set
S := {u < 1}, and assume that u(0) = 0, B1/K ⊂ S ⊂ BK , and that ∇u(S) ⊂⊂ Dom c-exp. Let
f, g be two densities such that
(5.1)
kf /λ1 − 1kC 0 (S) + kg/λ2 − 1kC 0 (Tu (S)) ≤ ε
for some constants λ1 , λ2 ∈ (1/2, 2) and ε ∈ (0, 1/4), and assume that (Tu )] f = g. Furthermore,
suppose that
(5.2)
kc + x · ykC 2 (BK ×BK ) ≤ δ.
Then there exist a universal constant γ ∈ (0, 1), and δ1 = δ1 (K) > 0 small, such that the following
holds: Let v be the solution of
(
det(D2 v) = λ1 /λ2
in Nδγ (co[S]),
v=1
on ∂ Nδγ (co[S]) .
Then
(5.3)
ku − vkC 0 (S) ≤ CK ε + δ γ/n
provided δ ≤ δ1 ,
4A similar result for the case c(x, y) = |x − y|p appeared in [7, Theorem 6.2]. Here, however, we have to deal with
some additional difficulties due to the fact that the c-exponential map is not necessarily defined on the whole Rn .
PARTIAL REGULARITY FOR OPTIMAL TRANSPORT MAPS
19
where CK is a constant independent of λ1 , λ2 , ε, and δ (but which depends on K).
Proof. First of all we observe that, since u(0) = 0, u = 1 on ∂S, S ⊂ BK , and kc + x · ykC 2 (BK ) ≤
δ 1, it is easy to check that there exists a universal constant a1 > 0 such that
(5.4)
|Dx c(x, y)| ≥ a1
∀ x ∈ ∂S, y = c-expx (∇u(x)).
Thanks to (5.4) and (5.2), it follows from the Implicit Function Theorem that, for each x ∈ ∂S,
the boundary of the set
Ex := z ∈ BK : c(z, y) − c(x, y) + u(x) ≤ 1
is of class C 2 inside BK , and its second fundamental form is bounded by CK δ, where CK > 0
depends only on K. Hence, since S can be written as
\
S :=
Ex ,
x∈∂S
it follows that
S is a (CK δ)-semiconvex set,
that is, for any couple of points x0 , x1 ∈ S the ball centered at x1/2 := (x0 + x1 )/2 of radius
0 δ (S) for some positive
CK δ|x1 − x0 |2 intersects S. Since S ⊂ BK , this implies that co[S] ⊂ NCK
0
constant CK depending only on K. Thus, for any γ ∈ (0, 1) we obtain
0 )δ γ (S).
Nδγ (co[S]) ⊂ N(1+CK
Since v = 1 on ∂ Nδγ (co[S]) and λ1 /λ2 ∈ (1/4, 4), by standard interior estimates for solution of
the Monge-Ampère equation with constant right hand side (see for instance [8, Lemma 1.1]), we
obtain
(5.5)
00
oscS v ≤ CK
(5.6)
00 δ γ/n ≤ v < 1
1 − CK
(5.7)
D2 v
≥
on ∂S,
00
in co[S],
Id /CK
00 depending only on
CK
δ γ/τ
K.
for some τ > 0 universal, and some constant
Let us define
√
√
√
√
00 γ/n
v + := (1 + 4ε + 2 δ)v − 4ε − 2 δ,
v − := (1 − 4ε − δ/2)v + 4ε + δ/2 + 2CK
δ .
Our goal is to show that we can choose γ universally small so that v − ≥ u ≥ v + on S. Indeed, if
we can do so, then by (5.5) this will imply (5.3), concluding the proof.
First of all notice that, thanks to (5.6), v − > u > v + on ∂S. Let us show first that v + ≤ v.
Assume by contradiction this is not the case. Then, since u > v + on ∂S,
∅=
6 Z := u < v + ⊂⊂ S.
Since v + is convex, taking any supporting plane to v + at x ∈ Z, moving it down and then lifting
it up until it touches u from below, we deduce that
(5.8)
∇v + (Z) ⊂ ∇u(Z)
(recall that both u and v + are of class C 1 ), thus by Remark 4.4
(5.9)
| c-exp(∇v + (Z))| ≤ |Tu (Z)|.
20
G. DE PHILIPPIS AND A. FIGALLI
We show that this is impossible. For this, using (5.7) and choosing γ := τ /4, for any x ∈ Z we
compute
√
√
D2 v + (x) + Dxx c x, c-expx (∇v + (x)) ≥ (1 + δ + 4ε)D2 v + δD2 v − δ Id
√
00
− δ) Id
≥ (1 + δ + 4ε)D2 v + (δ 3/4 /CK
√
≥ (1 + δ + 4ε)D2 v,
provided δ is sufficiently small, the smallness depending only on K. Thus, thanks (5.2) we have
√
det (1 + δ + 4ε)D2 v
det D2 v + (x) + Dxx c(x, c-expx (∇v + (x)))
≥
det Dxy c x, c-expx (∇v + (x)) 1+δ
√
λ1
≥ (1 + δ + 4ε)n (1 − 2δ)
λ2
λ1
≥ (1 + 4nε) .
λ2
In addition, thanks (5.7) and (5.2), since δ γ/τ = δ 1/4 δ we see that
D2 v + > kDxx ckC 0 (BK ×BK ) Id
inside co[S].
Hence, for any x, z ∈ Z, x 6= z and y = c-expx (∇v + (x)) (notice that c-expx (∇v + (x)) is well-defined
because of (5.8) and the assumption ∇u(S) ⊂⊂ Dom c-exp), it follows
Z
1 1 2 +
v + (z) + c(z, y) ≥ v + (x) + c(x, y) +
D v tz + (1 − t)x + Dxx c tz + (1 − t)x, y [z − x, z − x] dt
2 0
+
> v (x) + c(x, y),
where we used that ∇v + (x) + Dx c(x, y) = 0. This means that the supporting function z 7→
−c(z, y) + c(x, y) + v + (x) can only touch v + from below at x, which implies that the map Z 3 x 7→
c-expx (∇v + (x)) is injective. Thus, by Lemma 5.1 we get
λ1
(5.10)
| c-exp(∇v + (Z))| ≥ (1 + 4nε) |Z|.
λ2
On the other hand, since u is C 1 , it follows from (Tu )] f = g and (5.1) that
Z
λ1 (1 + ε)
λ1
f (x)
|Tu (Z)| =
dx ≤
|Z| ≤ (1 + 3ε) |Z|.
g(T
(x))
λ
(1
−
ε)
λ2
u
2
Z
This estimate combined with (5.10) shows that (5.9) is impossible unless Z is empty. This proves
that v + ≤ u.
The proof of the inequality v − ≤ u follows by the same argument except for a minor modification.
More precisely, let us assume by contradiction that W := {u > v − } is nonempty. In order to apply
the previous argument we would need to know that ∇v − (W ) ⊂ Dom c-exp. However, since the
gradient of v can be very large near ∂S, this may be a problem.
To circumvent this issue we argue as follows: since W is nonempty, there exists a positive
constant µ̄ such that u touches v − + µ̄ from below inside S. Let E be the contact set, i.e.,
E := {u = v − + µ̄}. Since both u and v − are C 1 , ∇u = ∇v − on E. Thus, if η > 0 is small
enough, then the set Wη := {u > v − + µ̄ − η} is nonempty and ∇v − (Wη ) is contained in a small
neighborhood of ∇u(Wη ), which is compactly contained in Dom c-exp. At this point, one argues
exactly as in the first part of the proof, with Wη in place of Z, to find a contradiction.
PARTIAL REGULARITY FOR OPTIMAL TRANSPORT MAPS
21
Theorem 5.3. Let u, f, g, η0 , δ0 be as in Theorem 4.3, and assume in addition that c ∈ C k,α (B3 ×
B3 ) and f, g ∈ C k,α (B1/3 ) for some k ≥ 0 and α ∈ (0, 1). There exist small constants η1 ≤ η0 and
δ1 ≤ δ0 such that, if
(5.11)
kf − 1C1 k∞ + kg − 1C2 k∞ ≤ δ1 ,
(5.12)
kc(x, y) + x · ykC 2 (B3 ×B3 ) ≤ δ1 ,
and
u − 1 |x|2 ≤ η1 ,
0
2
C (B3 )
(5.13)
then u ∈ C k+2,α (B1/9 ).
Proof. We divide the proof in two steps.
• Step 1: C 1,1 regularity. Fix a point x0 ∈ B1/8 , and set y0 := c-expx0 (∇u(x0 )). Up to replace u
(resp. c) with the function u1 (resp. c1 ) constructed in Steps 4 and 5 in the proof of Theorem 4.3,
we can assume that u ≥ 0, u(0) = 0, that
Sh := S(0, 0, u, h) = {u ≤ h},
and that
Dxy c(0, 0) = − Id .
(5.14)
Under these assumptions we will show that the sections of u are of “good shape”, i.e.,
(5.15)
B√h/K ⊂ Sh ⊂ BK √h
∀ h ≤ h1 ,
for some universal h1 and K. Arguing as in Step 6 of Theorem 4.3, this will give that u is C 1,1 at
the origin, and thus at every point in B1/8 .
First of all notice that, thanks to (5.13), for any h1 > 0 we can choose η1 = η1 (h1 ) > 0 small
enough such that (5.15) holds for Sh1 with K = 2. Hence, assuming without loss of generality that
δ1 ≤ 1, we see that
B√h1 /3 ⊂ Nδγ √h1 (co[Sh1 ]) ⊂ B3√h1 ,
1
where γ is the exponent from Proposition 5.2. Let v1 solve the Monge-Ampère equation
(
det(D2 v1 ) = f (0)/g(0) in Nδγ √h1 (co[Sh1 ]),
1
v 1 = h1
on ∂Nδγ √h1 (co[Sh1 ]).
1
√
Since B1/3 ⊂ Nδγ √h1 (co[Sh1 ])/ h1 ⊂ B3 , by standard Pogorelov estimates applied to the function
1
√
v1 ( h1 x)/h1 (see for instance [26, Theorem 4.2.1]), it follows that |D2 v1 (0)| ≤ M , with M > 0
some large universal constant.
Let hk := h1 2−k and define K̄ ≥ 3 to be the largest number such that any solution w of
det(D2 w) = f (0)/g(0) in Z,
(5.16)
with B1/K̄ ⊂ Z ⊂ BK̄ ,
w=1
on ∂Z,
22
G. DE PHILIPPIS AND A. FIGALLI
satisfies |D2 w(0)| ≤ M + 1. 5 We prove by induction that (5.15) holds with K = K̄.
If h = h1 then we already know that (5.15) holds with K = 2 (and so with K = K̄).
Assume now that (5.15) holds with h = hk and K = K̄, and we want to show that it holds with
h = hk+1 . For this, for any k ∈ N we consider uk the solution of
(
det(D2 vk ) = f (0)/g(0) in Nδγ √hk (co[Shk ]),
k
vk = h1 2−k
on ∂Nδγ √hk (co[Shk ]),
k
where
δk := kc(x, y) + x · ykC 2 (Sh
Let us consider the rescaled functions
p
ūk (x) := u hk x /hk ,
k
×Tu (Shk ))
v̄k (x) := vk
≤ δ1 .
p
hk x /hk .
Since by the inductive hypothesis B1/K̄ ⊂ S̄k := {ūk ≤ 1} ⊂ BK̄ , we can apply Proposition 5.2 to
deduce that
γ/n
γ/n
(5.17)
kūk − v̄k kC 0 (S̄k ) ≤ CK̄ osc f + osc g + δk
≤ CK̄ (δ1 + δ1 ).
Shk
Tu (Shk )
This implies in particular that, if δ1 is sufficiently small, B1/(2K̄) ⊂ {v̄k ≤ 1} ⊂ B2K̄ . By standard
estimates on the sections of solutions to the Monge-Ampère equation, the shapes of {v̄k ≤ 1} and
{v̄k ≤ 1/2} are comparable, and in addition sections are well included into each other [26, Theorem
3.3.8]: there exists a universal constant L > 1 such that
B1/(LK̄) ⊂ {v̄k ≤ 1/2} ⊂ BLK̄ ,
dist {v̄k ≤ 1/4}, ∂{v̄k ≤ 1/2} ≥ 1/(LK).
Using again (5.17) we deduce that, if δ1 is sufficiently small,
B1/(2LK̄) ⊂ {ūk ≤ 1/2} ⊂ B2LK̄ ,
so, by scaling back,
(5.18)
B√
hk+1 /(2LK̄)
dist {ūk ≤ 1/4}, ∂{ūk ≤ 1/2} ≥ 1/(2LK)
⊂ Shk+1 ⊂ B2LK̄ √h
k+1
,
p
dist Shk+2 , ∂Shk+1 ≥ hk /(2LK).
This allows us to apply Proposition 5.2 also to ūk+1 to get
(5.19)
kūk+1 − v̄k+1 kC 0 (S̄k+1 ) ≤ C2LK̄ osc f +
Shk+1
osc
Tu (Shk+1 )
γ/n
g + δk+1 .
We now observe that, by (5.15) and the C 1,β regularity of u (see Theorem 4.3), it follows that
β/2
diam(Shk ) + diam(Tu (Shk )) ≤ Chk ,
so by the C 0,α regularity of f and g, and the C 2,α regularity of c, we have (recall that γ < 1)
αβγ
γ/n
(5.20)
osc f + osc g + δk ≤ C 0 hσk ,
σ :=
Shk
2n
Tu (Shk )
5The fact that K̄ is well defined (i.e., 3 ≤ K̄ < ∞) follows by the following facts: first of all, by definition, M is
an a-priori bound for |D2 w(0)| whenever w is a solution of (5.16) with B1/3 ⊂ Z ⊂ B3 , so K̄ ≥ 3. On the other hand
p
K̄ ≤ 2(M + 1). Indeed, since 1/2 ≤ f (0)/g(0) ≤ 2 (by (5.11)) and M ≥ 1, the function
f (0) x22
+ x23 + . . . + x2n
g(0) M + 1
⊂ B1/√M +1 ⊂ {w̄ ≤ 1} ⊂ B√2(M +1) and |D2 w̄(0)| = 2(M + 1) .
w̄ := (M + 1)x21 +
is a solution of (5.16) such that B1/√2(M +1)
PARTIAL REGULARITY FOR OPTIMAL TRANSPORT MAPS
23
Hence, by (5.17) and (5.19),
kūk − v̄k kC 0 (S̄k ) + kūk+1 − v̄k+1 kC 0 (S̄k+1 ) ≤ C (CK̄ + C2LK̄ ) hσk ,
from which we deduce (recall that hk = 2hk+1 )
kvk − vk+1 kC 0 (Sh
k+1
)
≤ kvk − ukC 0 (Sh
k
+ ku − vk+1 kC 0 (Sh
)
k+1
)
= hk kūk − v̄k kC 0 (Sk ) + hk+1 kūk+1 − v̄k+1 kC 0 (Sk+1 )
≤ C (CK̄ + C2LK̄ ) h1+σ
.
k
Since vk and vk+1 are two strictly convex solutions of the Monge Ampère equation with constant
right hand side inside Shk+1 , and since Shk+2 is “well contained” inside Shk+1 , by classical Pogorelov
and Schauder estimates we get
(5.21)
(5.22)
kD2 vk − D2 vk+1 kC 0 (Sh
kD3 vk − D3 vk+1 kC 0 (Sh
k+2
)
0 σ
≤ CK̄
hk
σ−1/2
k+2
)
0
≤ CK̄
hk
,
0 is some constant depending only on K̄. By (5.21) applied to v for all j = 1, . . . , k (this
where CK̄
j
can be done since, by the inductive assumption, (5.15) holds for h = hj with j = 1, . . . , k) we
obtain
|D2 vk+1 (0)| ≤ |D2 v1 (0)| +
k
X
|D2 vj (0) − D2 vj+1 (0)|
j=1
≤M+
≤M+
0 σ
CK̄
h1
1
k
X
2−jσ
j=0
0
CK̄
hσ
− 2−σ 1
≤ M + 1,
provided we choose h1 small enough (recall that hk = h1 2−k ). By the definition of K̄ it follows
that also Shk+1 satisfies (5.15), concluding the proof of the inductive step.
• Step 2: higher regularity. Now that we know that u ∈ C 1,1 (B1/8 ), Equation (2.10) becomes
0
2,σ
uniformly elliptic. So one may use Evans-Krylov Theorem to obtain that u ∈ Cloc
(B1/9 ) for some
0
σ > 0, and then standard Schauder estimates to conclude the proof. However, for the convenience
0
of the reader, we show here how to give a simple direct proof of the C 2,σ regularity of u with
σ 0 = 2σ.
0
As in the previous step, it suffices to show that u is C 2,σ at the origin, and for this we have to
prove that there exists a sequence of paraboloids Pk such that
(5.23)
k(2+σ 0 )
sup |u − Pk | ≤ Cr0
Brk /C
0
for some r0 , C > 0.
Let vk be as in the previous step, and let Pk be their second order Taylor expansion at 0:
1
Pk (x) = vk (0) + ∇vk (0) · x + D2 vk (0)x · x.
2
24
G. DE PHILIPPIS AND A. FIGALLI
We observe that, thanks to (5.15),
(5.24)
kvk − Pk kC 0 (B(0,√h
k+2 /K))
≤ kvk − Pk kC 0 (Sh
k+2
)
≤ CkD3 vk kC 0 (Sh
k+2
3/2
) hk .
In addition, by (5.22) applied with j = 1, . . . , k and recalling that hk = h1 2−k and 2σ < 1 (see
(5.20)), we get
kD3 vk kC 0 (Sh
k+2
)
≤ kD3 v1 kC 0 (Sh ) +
k
X
3
kD3 vj − D3 vj+1 kC 0 (Sh
j+2
)
j=1
(5.25)
k
X
(σ−1/2)
σ−1/2
≤C 1+
hj
≤ Chk
.
j=1
Combining (5.15), (5.24), (5.25), and recalling (5.17) and (5.20), we obtain
ku − Pk kC 0 (B√h
k+2 /K
)
≤ kvk − Pk kC 0 (Sh
k+2
)
+ kvk − ukC 0 (Sh
k+2
)
≤ Ch1+σ
,
k
√
so (5.23) follows with r0 = 1/ 2 and σ 0 = 2σ.
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Scuola Normale Superiore, p.za dei Cavalieri 7, I-56126 Pisa, Italy
E-mail address: guido.dephilippis@sns.it
Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin
TX 78712, USA
E-mail address: figalli@math.utexas.edu